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A circular area with a radius of 6.50 $\mathrm{cm}$ lies in the $x$ -y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field $B=0.230 \mathrm{T}$ that points (a) in the $+z$ direction? (b) at an angle of $53.1^{\circ}$ from the $+z$ direction? (c) in the $+y$ direction?

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Physics 102 Electricity and Magnetism

Chapter 21

Electromagnetic Induction

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Magnetic Field and Magnetic Forces

Sources of Magnetic field

Inductance

Alternating Current

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Simon Fraser University

University of Winnipeg

Lectures

03:27

Electromagnetic induction is the production of an electromotive force (emf) across a conductor due to its dynamic interaction with a magnetic field. Michael Faraday is generally credited with the discovery of electromagnetic induction in 1831.

08:42

In physics, a magnetic field is a vector field that describes the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter (usually in the cgs system of units) and B is measured in teslas (SI units).

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for this problem. We have a circular region sitting in the X Y plane. In this setting, we have a uniform magnetic Byfield of 0.23 tests and the different parts. We're gonna be asked to find the flux at different angles of this be field. So let's recall that the magnetic flux in this uniforms setting was given by the perpendicular component of our Byfield multiplying by our area. He's also be a co sign fee where that fee is the angle between our be field and a perpendicular of our area. I'll draw that perpendicular area with this little red and back there. And since we have a circular region, we can write that area as pi r squared. All right, so for the first part, we're told that the Byfield is along the pluses e direction. I will draw that on here, and I'm just going to draw one back too. But since this is a uniform magnetic field, you can think about that area being everywhere in space. Anywhere we have that same magnetic fields to just think about that, that vector applying to anywhere in space. We have the angle between those two factors zero. And now we can plug everything into her a question. So we have 0.23 Tesla. Uh, our radius is 6.5 centimeters. You're gonna want to put that into meters squared. We have co sign zero, which is one this. If you plug it in, we get 0.0 305 levers. We were is being the unit of magnetic flux or a little bit nicer. And this is 3.5 milliliters. Great. So the next part we have our Byfield is 53.1 degrees from fussy drawing. That on Syrian looks something like this. And now we have the angle between those two vectors is 53.1 degrees. So plugging this in? Yeah, 0.23 huh? R squared and now co sign of 53.1 degrees. Quitting this in, we get 0.0 one a three levers or 1.83 milliliters. Lastly, we have our magnetic field pointed along the plus. Why direction? So that is Ah, there. Now we see that the angle between those two factors is 90 degrees. We know that 90 degrees has co signed 90 0 So we have no magnetic flux through this setting, and that makes sense. Flux weaken qualitatively. Think about the amount of stuff going through an area region here, all of those lines air pointing parallel to the face of that disk, the circular region. So we have no magnetic flux whatsoever. And this shows you if you have a uniform magnetic Byfield, really, it just comes down to finding the angle between you're be field and that perpendicular of the area, provided your area is flat throughout.

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